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<h1 class="reftitle">affineMap</h1>
<h2>Purpose</h2>
<p>Compute the affine map of the Polyhedron.</p>
<h2>Syntax</h2>
<pre class="synopsis">Q = P.affineMap(T)</pre>
<pre class="synopsis">Q = P.affineMap(T,method)</pre>
<pre class="synopsis">Q = affineMap(P,T,method)</pre>
<h2>Description</h2>
<p></p>
      Computes an affine map <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap1.png"> of polyhedron <tt>P</tt> to polyhedron <tt>Q</tt> 
      based on the transformation matrix <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap2.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap2.png">. The polyhedron <tt>Q</tt> is given by
    <p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap6.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap6.png"></p>    
    The matrix <tt>T</tt> must be real with <tt>n</tt> rows and <tt>d</tt> columns.
    <ul>
        
         <li>If <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap3.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap3.png"> then this operation is referred to as <em>projection</em>.</li>
        
         <li>If <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap4.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap4.png"> then this operation is referred to as <em>rotation/skew</em>.</li>
        
         <li>If <img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap5.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap5.png"> then this operation is referred to as <em>lifting</em>.</li>
    
      </ul>
  
   <h2>Input Arguments</h2>
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<td><tt>P</tt></td>
<td>
<p></p>Polyhedron in any format.<p>
	    		Class: <tt>Polyhedron</tt></p>
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<td><tt>T</tt></td>
<td>
<p></p>Transformation matrix.<p>
	    		Class: <tt>double</tt></p>
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<td><tt>method</tt></td>
<td>
<p></p>Specific method to use in projection operation. Allowed methods are "vrep", "fourier", and "mplp". 
            For details type "help Polyhedron/projection".<p>
	    		Class: <tt>string</tt></p>
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<h2>Output Arguments</h2>
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<td><tt>Q</tt></td>
<td>
<p></p>Polyhedron representing the affine map in H- or V-representation.<p>
	    		Class: <tt>Polyhedron</tt></p>
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<h2>Example(s)</h2>
<h3>Example 
				1</h3>Projection of the rectangle described in 2D.Define the rectangle <tt>P</tt> in V-representation.<pre class="programlisting">P = Polyhedron([0 0; 5 0; 5 3; 0 3]);</pre>
<pre class="programlisting"></pre>Compute the affine map of the rectangle with the matrix [-1 0.5] <pre class="programlisting">Q = P.affineMap([-1 0.5]) </pre>
<pre class="programlisting">Polyhedron in R^1 with representations:
    H-rep               : Unknown (call computeHRep() to compute)
    V-rep (redundant)   : Vertices   4 | Rays   0
Functions : none
</pre>We can see that <tt>Q</tt> is in dimension 1 while <tt>P</tt> is in dimension 2.<pre class="programlisting">plot([P,Q],'LineWidth',3); axis([-6 6 -1 4]); </pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap_img_1.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap_img_1.png" width="60%"></p>
<h3>Example 
				2</h3>Rotation of the rectangle described in 2D.Compute the affine map of the rectangle with the matrix [-1 0.5; -2 3] <pre class="programlisting">R = P.affineMap([-1 0.5; -2 3]) </pre>
<pre class="programlisting">Polyhedron in R^2 with representations:
    H-rep               : Unknown (call computeHRep() to compute)
    V-rep (redundant)   : Vertices   4 | Rays   0
Functions : none
</pre>We can see that <tt>Q</tt> remains in dimension 2.<pre class="programlisting">plot([P,R]); </pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap_img_2.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap_img_2.png" width="60%"></p>
<h3>Example 
				3</h3>Lifting of the rectangle described in 2D.Compute the affine map of the rectangle with the matrix [-1 0.5; -2 3; 0.8 -1.4] <pre class="programlisting">S = P.affineMap([-1 0.5; -2 3;-4 5]) </pre>
<pre class="programlisting">Polyhedron in R^3 with representations:
    H-rep               : Unknown (call computeHRep() to compute)
    V-rep (redundant)   : Vertices   4 | Rays   0
Functions : none
</pre>We can see that <tt>S</tt> is in dimension 3.<pre class="programlisting">plot([P,S]); </pre>
<pre class="programlisting"></pre>
<p class="programlistingindent"><img src="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap_img_3.png" alt="../../../../../../fig/mpt/modules/geometry/sets/@Polyhedron/affinemap_img_3.png" width="60%"></p>
<h2>See Also</h2>
<a href="./projection.html">projection</a>, <a href="../@ConvexSet/affinehull.html">affinehull</a><p></p>
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<br><p>©  <b>2010-2013</b>     Colin Neil Jones: EPF Lausanne,    <a href="mailto:colin.jones@epfl.ch">colin.jones@epfl.ch</a></p>
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